oof. In view of (11) it is sufficient to prove that (1) holds.
Consider
x∗y
∗
x∗z
∗
z∗y
= x∗
x∗z
∗y
∗
z∗y
= x∗y
∗y
∗x∗z
∗y∗
z∗y (by (12))
= x∗y
∗y
∗
z∗y∗x∗z
∗y
= x∗z
∗y
∗x∗z
∗y (by (12))
= 0.
(3.1)
This completes the proof.
In view of Theorems 3.3 and 3.4 and the comments made between them, we adopt
the following definitions for BCH-algebras.
Definition 3.5. A BCH